scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals MORSE INEQUALITIES ON CERTAIN INFINITE 2-COMPLEXES

2007 ◽  
Vol 49 (2) ◽  
pp. 155-165 ◽  
Author(s):  
R. AYALA ◽  
L. M. FERNÁNDEZ ◽  
J. A. VILCHES
Keyword(s):  
Finite Number ◽  
Morse Function ◽  
Dimensional Case ◽  
Morse Inequalities ◽  
Discrete Morse Function

AbstractUsing the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

scholarly journals Expected sizes of Poisson–Delaunay mosaics and their discrete Morse functions

2017 ◽  
Vol 49 (3) ◽  
pp. 745-767 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Anton Nikitenko ◽  
Matthias Reitzner
Keyword(s):  
Point Process ◽  
Poisson Point Process ◽  
Probabilistic Models ◽  
Morse Function ◽  
Discrete Point ◽  
Expected Number ◽  
Low Dimensions ◽  
Morse Functions ◽  
Point Set ◽  
Discrete Morse Function

AbstractMapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝn, we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensionsn≤ 4.

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

2021 ◽  
pp. 1-38
Author(s):  
Xianzhe Dai ◽  
Junrong Yan
Keyword(s):  
Essential Role ◽  
Morse Function ◽  
Noncompact Manifold ◽  
Morse Inequalities ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Witten Laplacian ◽  
Relative Cohomology ◽  
Witten Deformation ◽  
Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

scholarly journals On Ramsey-Minimal Infinite Graphs

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Jordan Barrett ◽  
Valentino Vito
Keyword(s):  
Ramsey Theory ◽  
Infinite Graphs ◽  
Minimal Graph ◽  
Finite Graphs ◽  
Minimal Graphs ◽  
Finite Case ◽  
Ramsey Minimal Graph ◽  
General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

scholarly journals Umbilical Submanifolds and Morse Functions

1972 ◽  
Vol 48 ◽  
pp. 197-201 ◽  
Author(s):  
Katsumi Nomizu ◽  
Lucio Rodríguez
Keyword(s):  
Euclidean Space ◽  
Differentiable Function ◽  
Critical Points ◽  
Morse Function ◽  
Differentiable Manifold ◽  
Morse Functions

Let Mn be a differentiable manifold (of class C∞). By a Morse function on Mn we mean a differentiable function whose critical points are all non-degenerate. If f is an immersion of Mn into a Euclidean space Rm, we may obtain Morse functions on Mn in the following way.

scholarly journals A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

2013 ◽  
Vol 9 (17) ◽  
pp. 11-20
Author(s):  
Carlos Cadavid ◽  
Juan Diego Vélez
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Critical Points ◽  
Morse Function ◽  
Initial Condition ◽  
Morse Functions ◽  
Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

scholarly journals Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

2019 ◽  
Vol 11 (4) ◽  
pp. 72-79
Author(s):  
Anna Kravchenko ◽  
Sergiy Maksymenko
Keyword(s):  
Homotopy Type ◽  
Morse Function ◽  
Previous Article ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Reeb Graph ◽  
Natural Right ◽  
Morse Functions ◽  
Path Component ◽  
Unique Vertex

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M, R)$ be a Morse function, and $\Gamma$ be its Kronrod-Reeb graph.Denote by $O(f)={f o h | h \in D(M)}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $D(M)$ onC^{\infty}$, and by $S(f)={h\in D(M) | f o h = f }$ the coresponding stabilizer of this function.It is easy to show that each $h\in S(f)$ induces an automorphism of the graph $\Gamma$.Let $D_{id}(M)$ be the identity path component of $D(M)$, $S'(f) = S(f) \cap D_{id}(M)$ be the subgroup of $D_{id}(M)$ consisting of diffeomorphisms preserving $f$ and isotopic to identity map, and $G$ be the group of automorphisms of the Kronrod-Reeb graph induced by diffeomorphisms belonging to $S'(f)$. This group is one of key ingredients for calculating the homotopy type of the orbit $O(f)$. In the previous article the authors described the structure of groups $G$ for Morse functions on all orientable surfacesdistinct from $2$-torus and $2$-sphere.  The present paper is devoted to the case $M = S^2$. In this situation $\Gamma$ is always a tree, and therefore all elements of the group $G$ have a common fixed subtree $Fix(G)$, which may even consist of a unique vertex. Our main result calculates the groups $G$ for all Morse functions $f: S^2 \to R$ whose fixed subtree $Fix(G)$ consists of more than one point.

scholarly journals Min-Max Theory for Cell Complexes

2020 ◽  
Vol 27 (03) ◽  
pp. 447-454
Author(s):  
Lacey Johnson ◽  
Kevin Knudson
Keyword(s):  
Morse Function ◽  
Critical Values ◽  
Discrete Version ◽  
Mountain Pass Lemma ◽  
Smooth Functions ◽  
Alternate Proof ◽  
Cell Complexes ◽  
Lusternik Schnirelmann ◽  
Regular Cell Complex ◽  
Discrete Morse Function

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

scholarly journals An advance in infinite graph models for the analysis of transportation networks

2016 ◽  
Vol 26 (4) ◽  
pp. 855-869
Author(s):  
Martín Cera ◽  
Eugenio M. Fedriani
Keyword(s):  
Graph Theory ◽  
Transportation Networks ◽  
Finite Graph ◽  
Infinite Graph ◽  
Average Degree ◽  
Infinite Graphs ◽  
Graph Models ◽  
Dangerous Goods ◽  
Finite Case ◽  
Percolation Thresholds

Abstract This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.

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